论文标题
计算有限小组在可定向表面上的等效类别：动态调查
Computing equivalence classes of finite group actions on orientable surfaces: A dynamic survey
论文作者
论文摘要
本文重点介绍了有限群体在黎曼表面的拓扑等效性类别的分类。通过Riemann-Hurwitz的界限，只有许多有限的小组在可定向的表面$ \ Mathcal {S} _g $ g \ geq 2 $的封闭式表面上行动。在$ \ Mathcal {s} _g $上，一个组的每一个此类操作，一个人可以将基本组$γ=π（\ Mathcal {o}）关联到商ORBIFOLD $ \ \ MATHCAL {O} = \ MATHCAL {o} = \ MATHCAL {s} _g g \ g \ g. Fuchsian集团完全由Orbifold的签名决定。 Riemann存在定理将$ \ Mathrm {g} $在$ \ Mathcal {s} _g $上存在的动作的问题减少到一个纯粹的组理论问题，即决定是否存在平稳的表达，将fuchsian $γ汇总到Group $ f imake $ \ mathrm \ g} $ {g} $ {使用计算机代数系统，例如\ textsc {magma}或差距，以及小组的库，在固定小属$ g \ geq 2 $表面上生成所有有限组动作的生成几乎成为一个例行程序。困难的部分是确定有关拓扑等效性的这些行为的类别。为了实现这一目标，需要调查Fuchsian组的自动形态组对$ \ Mathcal {s} _g $的有限组动作集的行动，并使用相应的签名。在本文中，我们得出了有限群体在黎曼表面的拓扑等效性的几个结果。作为一个应用程序，我们得出了$ g \ leq 9 $的有限群体动作的完整列表，以拓扑等效性。可以在附录中找到该动作的摘要，对更多详细信息感兴趣的读者将转介给网页[22]。预计我们将能够将列表扩展到更高的属，刷新的部分结果在网页上可用。以下文本是论文的扩展版本[23]。
This paper focuses on the classification of classes of topological equivalence of finite group actions on Riemann surfaces. By the Riemann-Hurwitz bound, there are just finitely many groups that act conformally on a closed orientable surface $\mathcal{S}_g$ of genus $g\geq 2$. With each such action of a group $\mathrm{G}$ on $\mathcal{S}_g$ one can associate the fundamental group $Γ=π(\mathcal{O})$ of the quotient orbifold $\mathcal{O}=\mathcal{S}_g/\mathrm{G}$, isomorphic to a Fuchsian group determined completely by orbifold's signature. The Riemann existence theorem reduces the problem of the existence of an action of $\mathrm{G}$ on $\mathcal{S}_g$ to a purely group-theoretical problem of deciding whether there is an smooth epimorphism mapping the Fuchsian group $Γ$ onto the group $\mathrm{G}$. Using computer algebra systems such as \textsc{Magma} or GAP, together with the library of small groups, the generation of all finite group actions on a surface of fixed small genus $g\geq 2$ becomes almost a routine procedure. The difficult part is to determine the classes of these actions with respect to topological equivalence. To achieve this, one needs to investigate the action of the automorphism group of a Fuchsian group on the set of finite group actions on $\mathcal{S}_g$ with the corresponding signature. In this paper we derive several results on the topological equivalence of finite group actions on Riemann surfaces. As an application, we derive complete lists of finite group actions of genus $g\leq 9$ distinguished up to the topological equivalence. A summary of the actions can be found in Appendix, the reader interested in more details is referred to the web page [22]. It is expected that we will be able to extend the list to higher genera, refreshed partial results are available on the web page. The following text is an extended version of the paper [23].
